Normalized solution for Kirchhoff equation with upper critical exponent and mixed Choquard type nonlinearities

Abstract

In this paper, we consider the existence of normalized solution to the following Kirchhoff equation with mixed Choquard type nonlinearities: \begin{equation*} \begin{cases} -\left(a + b \int_{\mathbb{R}^3} |\nabla u|^2 \, dx\right) \Delta u - \lambda u = \mu |u|^{q-2} u + (I_\alpha * |u|^{\alpha + 3}) |u|^{\alpha +1} u, \quad x \in \mathbb{R}^3, \\ \int_{\mathbb{R}^3} u^2 \, dx = \rho^2, \end{cases} \end{equation*} where $a,b,\rho >0$, $\alpha \in \left(0, 3\right)$, $\frac{14}{3} < q < 6$ and $\lambda \in \mathbb{R}$ will arise as a Lagrange multiplier. The quantity $\alpha + 3$ here represents the upper critical exponent relevant to the Hardy-Littlewood-Sobolev inequality, and this exponent can be regarded as equivalent to the Sobolev critical exponent $2^*$. We generalize the results by Wang et al.(Discrete and Continuous Dynamical Systems, 2025), which focused on nonlinear Kirchhoff equations with combined nonlinearities when $2< q< \frac{10}{3}$. The primary challenge lies in the necessity for subtle energy estimates under the \(L^2\)-constraint to achieve compactness recovery. Meanwhile, we need to deal with the difficulties created by the two nonlocal terms appearing in the equation.